Simple Harmonic Motion Oscillatory motion under a restoring force proportional to the amount of displacement from equilibrium A restoring force is a force that tries to move the system back to equilibrium For a spring-mass system like the one on the right, the restoring force is the spring force Important because SHM is a good model to describe vibrations of a guitar string, vibrations of atoms in molecules, etc.
A -A
A x t = Acos ωt -A v t = ωasin ωt a t = ω. Acos ωt
Energy in SHM E 010 = 1 2 ka. = 1 2 mv. + 1 2 kx. v x = k m (A. x. ) x = 0, v = max v =>? = k m A. 0 = ωa
Displace mass to x = A and let go Fill in the blank: Net force and acceleration are toward which direction? Max velocity is at x = Net force at O is Overshoots and compresses spring to x = Net force and acceleration are toward Max KE and max velocity at point 0 (equilibrium) Max PE, Max force, max acceleration, KE = 0, v= 0 at point A and B
Displace mass to x = A and let go Fill in the blank: Net force and acceleration are toward which direction? _Left Max velocity is at x = _0 Net force at O is _0 N Overshoots and compresses spring to x = -A Net force and acceleration are toward _Right
Amplitude (A) magnitude of displacement from Period (T) seconds per cycle; T = 1/f Frequency (f) cycles per second f = 1/T Angular frequency (ω) ω = 2πf = 2π/T
Terms for periodic motion Amplitude (A) magnitude of displacement from Period (T) seconds per cycle; T = 1/f Frequency (f) cycles per second f = 1/T Angular speed (ω) ω = 2πf = 2π/T
SHM and Circular Motion If the amplitude of the mass s oscillation is equal to the radius of the object in circular motion, and angular speed of the object in SHM = angular speed of object in circular motion Their motions are identical
SHM and Circular Motion
Things to note Period and frequency don t depend on amplitude A, even though object is traveling farther with larger A Bigger A = larger restoring Force = higher average velocity
A 17.0 g mass on a 35 N/m spring is pulled 20 cm from equilibrium and released. What is the position of the mass at time t = 1.2s? x = Acos(ωt) ω = k/m ω = (35 N/m)/(0.017 kg) ω = 45.4 rad/s x = (.2 m)cos[(45.4 rad/s)(1.2 s)] x = -0.101 m or -10.1 cm
SHM of simple pendulum Mathematically deriving the pendulum equation
Small angle approximation If the angle is small (< 10 degrees), sinθ θ
What s a wave? A wave is a wiggle in time and space The source of a wave is a vibration Vibrations are wiggles in time Wave is essentially a traveling vibration Wave does not transfer matter, it transfers energy
Qualities of a wave
Wave Speed In a freight train, each car is 10 m long. If two cars roll by you every second, how fast is the train moving? v = d/t = 2x(10 m)/(1 s) = 20 m/s A wave has a wavelength of 10 m. If the frequency is 2 Hz, how fast is the wave traveling? v = λf = (10 m)(2 Hz) = 20 m/s
Speed of a light wave c = 3.0 x 10^8 m/s Speed of sound (in dry air at 20º C) v = 340 m/s Speed of sound in a vacuum (in space)? v = 0 m/s
All waves on the electromagnetic spectrum have a wave speed of 3.0 x 10^8 m/s, they differ in their wavelengths and frequencies.
Wave speed depends on medium Sound waves travel faster if air is warmer, and travel faster in solids Wave speed = elasticity of medium/inertia of medium For a wave traveling down a rope v = D E ( F G ) Tension is restoring force, so greater tension means the rope returns to equilibrium faster, so wave can travel faster.
Waves carry energy, not matter Correction from lecture today: Waves also carry information! The amplitude, frequency, wavelength, and wave speed are considered information Amplitude modulation (AM) and frequency modulation (FM) are ways to use the information of a radio wave to translate to a sound wave.
Reflection When a wave reaches a boundary, it reflects ( bounces ) off the boundary If it reflects off a fixed point, it also inverts Newton s 3 rd law: wave exerts an upward force on the support, support exerts downward force on rope (see simulation)